Evaluate the Machian volume integral and determine the true MOND interpolating function

Evaluate the volume integral in Sciama’s relativistic nonlinear integral formulation of general relativity at low accelerations and determine the true MOND interpolating function, including whether its form is primarily fixed by the adopted cosmological model rather than by the nonlinear completion of the field equations.

Background

The paper obtains a phenomenological interpolating function close to MOND’s simple function by integrating Machian inertial and gravitoelectric terms over an acceleration-dependent causal volume in a simplified constant-Hubble cosmology. The authors emphasize that a fully relativistic treatment could alter this result.

They discuss Sciama’s nonlinear, generally covariant integral formulation of general relativity, in which local inertial forces depend on volume and surface integrals over stress-energy contributions and a retarded Green function within a specified horizon. Although the surface contribution vanishes in nonempty relativistic FLRW models, the low-acceleration volume integral has not yet been evaluated. The authors conjecture that the eventual interpolating function will be determined mainly by the cosmological model.

References

The evaluation of the volume integral at low accelerations remains a subject of ongoing research, which might result in the true interpolating function in agreement with observational constraints. Naively introducing the factor of $2$ missing in the linearized approximation to ($\ref{eq5}$), and rescaling $G$, leaves the shape of the interpolating function unchanged. I therefore conjecture that the true function arises primarily from the adopted cosmological model than from the nonlinear completion of the field equations themselves.

— Is MOND an effect of cosmic frame-dragging à la Mach?  (2610.09987 - Uruena, 7 Oct 2026) in Section 3, Discussion